These manuscripts also expands and improves the existing results of properties such as the propagation of higher Lebesgue integrability, initiated by the work Gustaffson, and later by Wennberg in [1]}, [2]}, obtaining estimates for the collision operator in Banach spaces \(L^r({\mathbb {R}}^d)\)  for \(1 \le r < \infty \) . Lions [3]} introduced the breakthrough concept of gain of integrabilty, by meaning that it is possible to control the gain operator estimates for hard potentials in \(L^r({\mathbb {R}}^d)\)  weighted spaces having lower growth when compared to the corresponding one obtained by the loss operator. This property, which still requires a bounded angular transition rate functions, may be viewed as a smoothing effect obtained under special conditions of the potential and angular transition rates or collision kernels. Later by Toscani, Villani and Mouhot [4]}, [5]} used these Lions' results to obtain not only the propagation and Sobolev regularity in the classical sense for the space homogenous Boltzmann flows. In the last decade, Carneiro and the authors revisited the work of Gustaffson [6]}, [6]} that inspired the work of the authors [8]} to obtained significant improvements removing conditions from the work of Lions on the transition probability kernels, but also quantified the rate of integrability estimates
by parameters depending on the Cauchy problem data and the \(L^r({\mathbb {R}}^d)\)  norm of the initial data.
